Arithmetic or Geometric? a) If a_{1}, a_{2}, a_{3}, \cdots is an

usagirl007A

usagirl007A

Answered question

2021-08-13

Arithmetic or Geometric?
a) If a1,a2,a3, is an arithmetic sequence, is the sequence a1+2,a2+2,a3+2, arithmetic?
b) If a1,a2,a3, is a geometric sequence, is the sequence 5a1,5a2,5a3, geometric?

Answer & Explanation

Khribechy

Khribechy

Skilled2021-08-14Added 100 answers

a) To show: the sequence a1+2,a2+2,a3+2, is arithmetic.
Given:
The sequence a1,a2,a3, is an arithmetic sequence.
Approach:
In arithmetic sequence, difference between the terms remains constant throughout.
Calculation:
The sequence a1,a2,a3, is an arithmetic sequence.
a2a1=a3a2=(1)
(a2+2)(a1+2)=(a2a1)
(a3+2)(a2+2)=(a3a2)
From equation (1)
(a3+2)(a2+2)=(a2a1)
As the differences between the terms are same then it is an arithmetic sequence.
Conclusion: hence, the sequence a1+2,a2+2,a3+2, is arithmetic.
b) To show: 5a1,5a2,5a3, is a geometric sequence.
Given: The sequence a1,a2,a3, is geometric sequence.
Approach: In a geometric sequence each term is found by multiplying the previous term by a constant.
Calculation:
a2a1=a3a2=(1)
Multiplying equation (1) by 5
5a25a1=5a35a2=
Then it is a geometric sequence.
Conclusion: hence, the sequence 5a1,5a2,5a3, is a geometric sequence.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?